Propagation of the Wavefunction

The perturbed-cluster approach [701,702] is formulated in the LCAO approximation and is based on the following sequence of steps, [703] a) subdivide the entire defect system into a molecular cluster (C), containing the defect, and an external region (D), the indented crystal b) calculate the wavefunction for the molecular cluster in the field of the indented crystal c) correct the cluster solution in order to allow for the propagation of the wavefunction into the indented crystal while generating the density matrix of the defect system. Steps b and c are repeated to self-consistency. The corrective terms in step c are evaluated by assuming that the density of states projected onto the indented crystal is the same as in the perfect host crystal (fundamental approximation). 

On the other hand, the state (a or jS) is determined from the propagation of the wavefunction via equation (5) and this forms the basis of the trajectory surface hopping method that we shall presently discuss. 

The time propagation of the MCSCF electronic wavefunctions is achieved through the following representation of the orbital and configuration parameters ... 

The functions (r. R.O.t = 0) involve a product of the initial wavefunction and the internal coordinate dejjendent vector components of the transition dipole moment (see Ref. [43] and Appendix B of Ref. [33] ). As the total angular momentum is a conserved quantity during the time propagation of the wavepacket, we may divide up the initied wavepacket (Eq. (23)) into three components [43], one for each of the allowed valuers of J. Thus E(p (23) may be rewritten as ... 

For the propagation of the multichannel wavefunction 4>(R), in real or complex-scaled coordinates, an efficient algorithm is furnished by fhe Fox-Goodwin-Numerov method [8, 44], which results from a discrefizafion of the differential operator appearing in Eqs. (39). Given adjacent points R — h, R, and R + h on the grid, we define an inward mafrix (labeled /) and an outward matrix (labeled o) as ... 

We now describe our CASSCF implementation of the Ehrenfest method that allows the electronic wavefunction to be made of multiple states. An approximate second-order method with a numerical fit procedure is used for the propagation of the nuclei. 

The non-adiabatic couplings are, therefore, included in our propagation of the electronic wavefunction by finite... 

In the first calculation, we propagate only the electronic wavefunction without allowing the nuclei to move. This is to study first pure charge migration. In the second calculation, the nuclei are allowed to move along with the propagation of the electronic wavefunction. In Fig. 5, the spin densities partitioned onto the atoms are plotted as a function of time in each case. The bottom half aims to represent the corresponding evolution of both the... 

Bom-Oppenheimer molecular dynamics molecular dynamics with electronic stracture calculations on the fly, diagonalization in each step Car-ParrineUo molecular dynamics molecular dynamics with electronic stmcture calculations on the fly, orthogonaUzatiOTi in each step otherwise the coefficients of the wavefunction are propagated like the nuclear positions Collective variables... 

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